A001815 -id:A001815 - cp's OEIS frontend

A340257 a(n) = 2^n * (1+n*(n+1)/2).

1, 4, 16, 56, 176, 512, 1408, 3712, 9472, 23552, 57344, 137216, 323584, 753664, 1736704, 3964928, 8978432, 20185088, 45088768, 100139008, 221249536, 486539264, 1065353216, 2323644416, 5049942016, 10938744832, 23622320128, 50868518912, 109253230592, 234075717632

Offset: 0

Alois P. Heinz, Jan 02 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Partial sums of A080929.

Cf. A000079, A000124, A001787, A001815, A036289, A087431, A340298.

a:= n-> 2^n*(1+n*(n+1)/2):
seq(a(n), n=0..30);

Table[2^n (1+(n(n+1))/2),{n,0,30}] (* or *) LinearRecurrence[{6,-12,8},{1,4,16},30] (* Harvey P. Dale, Jan 19 2023 *)

G.f.: (4*x^2-2*x+1)/(1-2*x)^3.
E.g.f.: exp(2*x)*(2*x^2+2*x+1).
a(n) = A000079(n) + A001815(n+1).
a(n) = A000079(n) * A000124(n).
a(n) = 2*a(n-1) + n*2^n = 2*a(n-1) + A036289(n), assuming a(-1) = 1/2.
a(n) = A340298(2^n).
a(n) = 2 * A087431(n) for n > 0.
a(n) = 4 * A007466(n) for n > 0.

A052791 3^(n-3)n(n-1)*(n-2).

Original entry on oeis.org

0, 0, 0, 6, 72, 540, 3240, 17010, 81648, 367416, 1574640, 6495390, 25981560, 101328084, 386889048, 1450833930, 5356925280, 19514513520, 70252248672, 250273635894, 883318714920, 3091615502220, 10739295955080, 37050571045026, 127030529297232, 433058622604200, 1468633589701200

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

The number of surjective functions f:{1,2,...,n}->{1,2,3} with a designated pre-image of 1, 2, and 3.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 748
Index entries for linear recurrences with constant coefficients, signature (12,-54,108,-81).

Cf. A001815.

spec := [S,{B=Set(Z),S=Prod(Z,Z,Z,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Range[0, 20]! CoefficientList[Series[(x Exp[x])^3, {x, 0, 20}], x]
LinearRecurrence[{12,-54,108,-81},{0,0,0,6},30] (* Harvey P. Dale, Sep 02 2022 *)

a(n)=3^(n-3)*n*(n-1)*(n-2); /* Joerg Arndt, Sep 16 2012 */

E.g.f.: x^3*exp(x)^3
Recurrence: {a(1)=0, a(2)=0, a(3)=6, (-3*n-3)*a(n)+(-2+n)*a(n+1)}.
a(n) = n!*sum(i+j+k=n, ijk/(i!j!k!)) - Benoit Cloitre, Nov 11 2004
G.f.: 6*x^3 / (3*x-1)^4. - Colin Barker, Jun 04 2013

Edited by N. J. A. Sloane, Dec 24 2010

A089264 Number of permutations of length n containing exactly once 132 and 213, likewise for pattern pair (231,312).

Original entry on oeis.org

3, 6, 17, 42, 102, 242, 564, 1296, 2944, 6624, 14784, 32768, 72192, 158208, 345088, 749568, 1622016, 3497984, 7520256, 16121856, 34471936, 73531392, 156499968, 332398592, 704643072, 1491075072, 3149922304, 6643777536, 13992198144

Offset: 4

Ralf Stephan, Oct 30 2003

Aaron Robertson, Permutations restricted by two distinct patterns of length three, arXiv:math/0012029 [math.CO], 2000.
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A001815.

LinearRecurrence[{6,-12,8},{3,6,17,42,102,242},40] (* Harvey P. Dale, Apr 10 2022 *)

For n>=7, a(n) = (n^2+21*n-28)*2^(n-9).

G.f.: x^4*(x-1)^2*(2*x^3-2*x^2+6*x-3) / (2*x-1)^3. [Colin Barker, Jan 31 2013]

A224288 Number of permutations of length n containing exactly 2 occurrences of 123 and 2 occurrences of 132.

Original entry on oeis.org

0, 0, 0, 0, 1, 6, 26, 94, 306, 934, 2732, 7752, 21488, 58432, 156288, 411904, 1071104, 2750976, 6984704, 17545216, 43634688, 107511808, 262602752, 636223488, 1529741312, 3652059136, 8660975616, 20412104704, 47826599936, 111446851584, 258360737792, 596044152832

Offset: 0

Brian Nakamura, Apr 03 2013

			a(4) = 1: (1,2,4,3).
a(5) = 6: (2,3,5,1,4), (2,3,5,4,1), (2,5,1,3,4), (3,1,4,5,2), (4,1,2,5,3), (5,1,2,4,3).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
B. Nakamura, Approaches for enumerating permutations with a prescribed number of occurrences of patterns, arXiv 1301.5080 [math.CO], 2013.
B. Nakamura, A Maple package for enumerating n-permutations with r occurrences of the pattern 123 and s occurrences of the pattern 132 [Broken link]
Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).

Cf. A000079, A001787, A001815, A046718, A001793.

# Programs can be obtained from the Nakamura link

Join[{0, 0, 0, 0, 1}, LinearRecurrence[{10, -40, 80, -80, 32}, {6, 26, 94, 306, 934}, 27]] (* Jean-François Alcover, Feb 29 2020 *)

G.f.: -(2*x^5+6*x^4-6*x^3+6*x^2-4*x+1)*x^4/(2*x-1)^5. - Alois P. Heinz, Apr 03 2013

a(n) = 2^(-11+n)*(1504-994*n+219*n^2-18*n^3+n^4) for n>4. - Colin Barker, Apr 14 2013

A224290 Number of permutations of length n containing exactly 3 occurrences of 123 and 3 occurrences of 132.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 6, 30, 136, 566, 2176, 7808, 26440, 85332, 264632, 793792, 2315136, 6592640, 18390784, 50392064, 135921664, 361536512, 949708800, 2466807808, 6342115328, 16153509888, 40790523904, 102186352640, 254105092096, 627533152256, 1539764125696

Offset: 0

Brian Nakamura, Apr 03 2013

nonn

			a(5) = 1: (1,4,3,2,5).
a(6) = 6: (2,5,4,3,1,6), (2,5,4,3,6,1), (3,5,1,4,6,2), (3,6,1,4,2,5), (5,1,4,3,2,6), (6,1,4,3,2,5).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
B. Nakamura, Approaches for enumerating permutations with a prescribed number of occurrences of patterns, arXiv 1301.5080 [math.CO], 2013.
B. Nakamura, A Maple package for enumerating n-permutations with r occurrences of the pattern 123 and s occurrences of the pattern 132 [Broken link]
Index entries for linear recurrences with constant coefficients, signature (14,-84,280,-560,672,-448,128).

Cf. A000079, A001787, A001815, A046718, A001793.

# Programs can be obtained from the Nakamura link

Join[{0, 0, 0, 0, 0, 1, 6}, LinearRecurrence[{14, -84, 280, -560, 672, -448, 128}, {30, 136, 566, 2176, 7808, 26440, 85332}, 33]] (* Jean-François Alcover, Nov 28 2018 *)

concat([0,0,0,0,0], Vec(x^5*(1 - 8*x + 30*x^2 - 60*x^3 + 62*x^4 - 36*x^5 + 24*x^6 - 8*x^7 + 4*x^8) / (1 - 2*x)^7 + O(x^40))) \\ Colin Barker, Nov 28 2018

G.f.: -(4*x^8-8*x^7+24*x^6-36*x^5+62*x^4-60*x^3+30*x^2-8*x+1)*x^5 / (2*x-1)^7. - Alois P. Heinz, Apr 03 2013
From Colin Barker, Nov 28 2018: (Start)
a(n) = (1/9)*2^(n-15) * (307008 - 247512*n + 78118*n^2 - 12087*n^3 + 937*n^4 - 33*n^5 + n^6) for n>6.
a(n) = 14*a(n-1) - 84*a(n-2) + 280*a(n-3) - 560*a(n-4) + 672*a(n-5) - 448*a(n-6) + 128*a(n-7) for n>13.
(End)

A377582 Expansion of e.g.f. (1 + x * exp(x))^3.

Original entry on oeis.org

1, 3, 12, 51, 228, 1035, 4698, 21063, 92424, 395091, 1643790, 6664383, 26387100, 102286587, 389125506, 1455994935, 5368721808, 19541252259, 70312410774, 250408115823, 883617559140, 3092276105163, 10740749281482, 37053754521831, 127037475064728, 433073722098675

Offset: 0

Seiichi Manyama, Nov 02 2024

Cf. A002999, A377583.
Cf. A001815, A052791.
Cf. A377530, A377575.

a(n) = n!*sum(k=0, n, k^(n-k)*binomial(3, k)/(n-k)!);

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3,k)/(n-k)!.

G.f.: (1-17*x+127*x^2-542*x^3+1453*x^4-2543*x^5+2973*x^6-2478*x^7+1626*x^8-648*x^9) / ((1-x)^2*(1-2*x)^3*(1-3*x)^4).

A140880 Triangle read by rows, T(n,k) = Gamma(n+3)/(Gamma(k+1)*Gamma(n-k+1)) for n>=0 and 0<=k<=n.

Original entry on oeis.org

2, 6, 6, 12, 24, 12, 20, 60, 60, 20, 30, 120, 180, 120, 30, 42, 210, 420, 420, 210, 42, 56, 336, 840, 1120, 840, 336, 56, 72, 504, 1512, 2520, 2520, 1512, 504, 72, 90, 720, 2520, 5040, 6300, 5040, 2520, 720, 90

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jul 22 2008

			Triangle starts:
[0] 2
[1] 6,    6
[2] 12,  24,   12
[3] 20,  60,   60,   20
[4] 30, 120,  180,  120,   30
[5] 42, 210,  420,  420,  210,   42
[6] 56, 336,  840, 1120,  840,  336,   56
[7] 72, 504, 1512, 2520, 2520, 1512,  504,  72
[8] 90, 720, 2520, 5040, 6300, 5040, 2520, 720, 90

T(n,0) = T(n,n) = A002378(n+1). T(n,k) = 2*A094305(n,k).
Row sums are A001815(n+2).
Cf. A007318.

T := (n,k) -> GAMMA(n+3)/(GAMMA(k+1)*GAMMA(n-k+1)):
seq(seq(T(n,k), k=0..n), n=0..8); # Peter Luschny, Oct 29 2017

Flatten[Table[Gamma[n+3]/(Gamma[k+1]Gamma[n-k+1]),{n,0,8},{k,0,n}]]

Edited and new name from Peter Luschny, Oct 29 2017

A224289 Number of permutations of length n containing exactly 1 occurrence of 123 and 2 occurrences of 132.

Original entry on oeis.org

0, 0, 0, 2, 8, 26, 79, 232, 664, 1856, 5072, 13568, 35584, 91648, 232192, 579584, 1427456, 3473408, 8359936, 19922944, 47054848, 110231552, 256311296, 591921152, 1358430208, 3099590656, 7034896384, 15888023552, 35718692864, 79960211456, 178291474432, 396076515328, 876844417024

Offset: 1

Brian Nakamura, Apr 03 2013

B. Nakamura, Approaches for enumerating permutations with a prescribed number of occurrences of patterns, arXiv 1301.5080, 2013.
Index entries for linear recurrences with constant coefficients, signature (8, -24, 32, -16).

Cf. A000079, A001787, A001815, A046718, A001793.

# Programs can be obtained from author's personal website.

LinearRecurrence[{8,-24,32,-16},{0,0,0,2,8,26,79},40] (* Harvey P. Dale, Jun 23 2017 *)

a(n) = 2^(-8+n)*(-136+70*n-11*n^2+n^3) for n>3. G.f.: -x^4*(x^3-10*x^2+8*x-2) / (2*x-1)^4. - Colin Barker, Apr 14 2013

A224291 Number of permutations of length n containing exactly 4 occurrences of 123 and 4 occurrences of 132.

Original entry on oeis.org

0, 0, 0, 0, 1, 11, 60, 270, 1084, 4028, 14144, 47577, 154740, 489728, 1514786, 4593118, 13682374, 40106060, 115824376, 329901232, 927585696, 2576685888, 7076644480, 19228648192, 51725149184

Offset: 1

Brian Nakamura, Apr 03 2013

nonn

B. Nakamura, Approaches for enumerating permutations with a prescribed number of occurrences of patterns, arXiv 1301.5080, 2013.

Cf. A000079, A001787, A001815, A046718, A001793.

# Programs can be obtained from author's personal website.

A370469 Triangle read by columns where T(n,k) is the number of points in Z^n such that |x1| + ... + |xn| = k, |x1|, ..., |xn| > 0.

Original entry on oeis.org

2, 2, 4, 2, 8, 8, 2, 12, 24, 16, 2, 16, 48, 64, 32, 2, 20, 80, 160, 160, 64, 2, 24, 120, 320, 480, 384, 128, 2, 28, 168, 560, 1120, 1344, 896, 256, 2, 32, 224, 896, 2240, 3584, 3584, 2048, 512, 2, 36, 288, 1344, 4032, 8064, 10752, 9216, 4608, 1024

Offset: 1

Shel Kaphan, Mar 30 2024

T(n,k) is the number of points on the n-dimensional cross polytope with facets at distance k from the origin which have no coordinate equal to 0.
T(n,n) = 2^n. The (n-1)-dimensional simplex at distance n from the origin in Z^n has exactly 1 point with no zero coordinates, at (1,1,...,1). There are 2^n (n-1)-dimensional simplexes at distance n from the origin as part of the cross polytope in Z^n. (The lower dimensional facets do not count as they have at least one 0 coordinate.)
T(2*n,3*n) = T(2*n+1,3*n), and this is A036909.

			 n\k 1 2 3  4  5   6   7    8    9    10    11    12     13     14      15
   -----------------------------------------------------------------------
 1 | 2 2 2  2  2   2   2    2    2     2     2     2      2      2       2
 2 |   4 8 12 16  20  24   28   32    36    40    44     48     52      56
 3 |     8 24 48  80 120  168  224   288   360   440    528    624     728
 4 |       16 64 160 320  560  896  1344  1920  2640   3520   4576    5824
 5 |          32 160 480 1120 2240  4032  6720 10560  15840  22880   32032
 6 |              64 384 1344 3584  8064 16128 29568  50688  82368  128128
 7 |                 128  896 3584 10752 26880 59136 118272 219648  384384
 8 |                      256 2048  9216 30720 84480 202752 439296  878592
 9 |                           512  4608 23040 84480 253440 658944 1537536
10 |                                1024 10240 56320 225280 732160 2050048
11 |                                      2048 22528 135168 585728 2050048
12 |                                            4096  49152 319488 1490944
13 |                                                   8192 106496  745472
14 |                                                         16384  229376
15 |                                                                 32768
The cross polytope in Z^3 (the octahedron) with points at distance 3 from the origin has 8 triangle facets, each with edge length 4. There is one point in the center of each triangle with coordinates (+-1,+-1,+-1).

Cf. A033996, A333714 (n=3)
Cf. A102860 (n=4).
Cf. A036289, A097064, A134401 (+1-diagonal).
Cf. A001815 (+2-diagonal).
Cf. A371064.
Cf. A036909.
2 * A013609.

T[n_,k_]:=Binomial[k-1,n-1]*2^n; Table[T[n,k],{k,10},{n,k}]//Flatten

from math import comb
def A370469_T(n,k): return comb(k-1,n-1)<Chai Wah Wu, Apr 25 2024

T(n,k) = binomial(k-1,n-1)*2^n.

G.f.: 2*x*y/(1 - y - 2*x*y). - Stefano Spezia, Apr 27 2024

cp's OEIS Frontend

A340257 a(n) = 2^n * (1+n*(n+1)/2).

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Maple

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A052791 3^(n-3)*n*(n-1)*(n-2).

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A089264 Number of permutations of length n containing exactly once 132 and 213, likewise for pattern pair (231,312).

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Programs

Mathematica

Formula

A224288 Number of permutations of length n containing exactly 2 occurrences of 123 and 2 occurrences of 132.

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Author

Keywords

Examples

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Crossrefs

Programs

Maple

Mathematica

Formula

A224290 Number of permutations of length n containing exactly 3 occurrences of 123 and 3 occurrences of 132.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A377582 Expansion of e.g.f. (1 + x * exp(x))^3.

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Author

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Crossrefs

Programs

PARI

Formula

A140880 Triangle read by rows, T(n,k) = Gamma(n+3)/(Gamma(k+1)*Gamma(n-k+1)) for n>=0 and 0<=k<=n.

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Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions

A224289 Number of permutations of length n containing exactly 1 occurrence of 123 and 2 occurrences of 132.

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Author

Keywords

Links

Crossrefs

A052791 3^(n-3)n(n-1)*(n-2).